Main

Over the previous decades, rice-breeding practices have greatly increased grain yield of rice cultivars, largely related to the adoption of semi-dwarf alleles and efficient use of hybrid vigour (heterosis)1,2. In hybrid rice varieties (Oryza sativa), the heterozygous first filial (F1) generation display a yield advantage of 10–20% over their inbred parental lines2,3,4, making heterosis breeding a powerful way to meet global food-security demands. The genomic basis of heterosis has been extensively studied in a number of crops, and explained by several classical models. Previous research of multiple crops has provided support for three non-mutually exclusive hypotheses of dominance, single-locus overdominance and pseudo-overdominance models for genomic loci contributing to heterosis5,6,7,8,9,10,11,12,13.

The genetic basis of heterosis for rice grain yield has remained largely uncharacterized. The second filial (F2) populations generated from the elite hybrids (F1) can provide useful information for heterosis analysis. Assuming a single polymorphic locus contributes to heterosis, it is expected that three genotypes (homozygous male-parent genotype MM, heterozygous genotype MF and homozygous female-parent genotype FF) will be present in F2 with a proportion of 1:2:1 at the local genomic region, allowing for an estimation of heterotic effects. Here we investigated F2 populations from 17 representative hybrid rice crosses, in order to identify loci contributing to yield traits, analyse their heterotic effects and evaluate their contributions to heterotic advantage.

Sequencing, genotyping and genetic mapping

The strategy of this study is briefly described in Extended Data Fig. 1. We have previously constructed a genome map for 1,495 elite hybrid rice varieties13. According to both genomic information and phenotypic performance of the hybrid rice varieties, 17 representative hybrid combinations were selected and the corresponding F1 hybrids were mostly important varieties of super-hybrid rice that were widely cultivated during recent years (Extended Data Fig. 2). We generated 17 sets of large recombinant populations. The resulting 10,074 F2 lines included 9 populations from O. sativa subspecies (ssp.) indica-indica crosses of a three-line system (cytoplasmic male sterility (CMS) for female parental lines, type A, n = 3,947), 6 populations from indica-indica crosses of a two-line system (environmentally sensitive male sterility, type B, n = 3,957), and 2 populations from O. sativa ssp. indica–O. sativa ssp. japonica crosses (type C, n = 2,170). We sequenced the genomes of the 10,074 F2 lines and constructed a dense genotype map with a total of 347,803 recombination events using a high-throughput sequence-based genotyping method14 (Fig. 1a and Extended Data Fig. 3). In the genotype map of each F2 population, the proportion of three genotypes (MM, MF and FF) agreed with expectation of 1:2:1 across the vast majority of the rice genome (Extended Data Fig. 3a).

Figure 1: Large-scale sequencing, genotyping and genetic mapping in 10,074 F2 lines.
figure 1

a, Plots of the first two principal components of 10,074 lines using whole-genome single-nucleotide polymorphisms (SNPs). b, The Manhattan plots of a genome-wide association study (GWAS) for heading date in 10,074 lines. Negative log10 P values from linear mixed model (y axis) are plotted against SNP positions (x axis) on each of 12 rice chromosomes. The genome-wide significant P-value threshold (10−6) is indicated by a horizontal dashed line. The loci with well-characterized genes are indicated near the association peaks. c, Manhattan plot for GWAS of plant height in 10,074 lines. The arrow indicates the newly identified loci for plant height.

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The yield traits were analysed using a composite interval-mapping method for each population separately to identify large-effect quantitative trait loci (QTLs) with resolution mostly within 300 kb. The alleles segregating in multiple populations could also be detected through genome-wide associations that combined information across the 17 populations (Fig. 1b, c). The QTLs included 23 candidate genes previously implicated in grain yield, and additional new QTLs for grain yield identified in this work. We were not able to resolve these QTLs to single-gene level, and additional studies will be needed to further map these and identify candidate genes.

For the 15 populations from indica–indica crosses, we identified 474 QTLs for 7 yield traits (Fig. 2a). We then examined the genotyping data for the local genomic region of each of these QTLs to measure the effect of three genotypes (MM, MF and FF) (Methods). The QTLs for grain-yield-related traits showed enrichment with positive dominance effects (the effect of heterozygous genotype was similar to that of homozygous advantageous alleles), especially for plant height and seed-setting rate, providing the basis to maintain or even increase the grain yield, in the case of heterozygous genomes (Fig. 2a).

Figure 2: Evaluation of dominance effects for all yield-related QTLs.
figure 2

a, Plots of the dominance effect for 474 QTLs (likelihood of odds (LOD) > 3.5) in the 15 populations underlying seven yield-related traits. b, The genomic locations of seven loci with strong phenotypic effects for grain yield per plant (LOD > 3.5) identified in the population from a cross between 93–11 and Y58S. The alleles with better yield performance that were contributed from the male parent 93–11 and the female parent Y58S are coloured by blue and red, respectively. c, Plots of the dominance effect and the allele effect for the seven QTLs. d, For all QTLs underlying grain yield per plant (LOD > 3.5) in the 15 populations from indica–indica crosses, the proportions of the advantageous alleles contributors (from male parents and female parents) are indicated.

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Furthermore, we found that, for most of the grain-yield QTLs, the alleles with higher yield performance were contributed from male parents (parental lines that restored fertility in hybrid rice) (Fig. 2b–d). As the heterozygous state of the QTLs mostly represented positive partial dominance, assuming no interaction between loci, the combined total of the effects of an advantageous allele from only the male parents at most insured that the F1 lines had a yield performance close to their male parents, but that cannot surpass the phenotypic value of that parent. Nevertheless, most hybrids showed a yield advantage of >10% over their male parents, suggesting that the allele from female parents also contributes to heterosis. Hence, we addressed the genetic contributors that led to hybrids outperforming their male parents for each type of hybrid system.

Type-A system: hd3a and tac1

The hybrid rice system of type A, developed in the 1970s, is the most common system, and includes over 75% of known hybrid varieties. For type-A-system hybrid lines, we identified heterosis QTLs to include candidate genes hd3a and tac1 through a combination of genomic mapping and allelic mutations in the coding regions (see Methods). For both candidate genes, the alleles in female parents influenced the yield performance of the hybrid. Hd3a is a rice orthologue of the Arabidopsis thaliana FT gene15. For its orthologue gene in tomato, yield overdominance from SFT heterozygosity has previously been observed5. The allele hd3a that was associated with heterosis had a two-base substitution, which is also associated with delay of flowering in long-day conditions (Fig. 3a). The estimation of allelic effects in type-A populations demonstrated that Hd3a also had large effect on grain yield. We further evaluated the effect of the three genotypes at the Hd3a locus. Compared to the genotype in male parents (Hd3a/Hd3a), the heterozygous state (Hd3a/hd3a) showed an advantage of 7.4% in seed-setting rate (varying from 2.2% to 10.9% in different hybrids) and 9.9% in grain yield per plant (varying from 1.4% to 17.9%) (Fig. 3b). We then examined the effect of the male parents with the hd3a/hd3a genotype on hybrid performance. We noticed that, although the hd3a/hd3a genotype showed an increased grain yield, it also showed delayed rice flowering (extending the growth stage by up to 20%), which would be impractical for agricultural production in most rice-planting areas (Extended Data Fig. 4). Fortunately, the heterozygous state (Hd3a/hd3a) showed an appropriate effect for rice heading: the heterozygous genotype only slightly delayed flowering time compared with the wild-type Hd3a/Hd3a. Hence, when considering both yield and timing, the performance of Hd3a/hd3a was better than either of the two homozygous genotypes.

Figure 3: Two candidate genes hd3a and tac1 for heterosis for yield traits in type-A hybrids.
figure 3

a, Gene structure of Hd3a and hd3a alleles. b, Plots of the advantage of the heterozygous state (Hd3a/hd3a) over Hd3a/Hd3a for grain yield per plant and yield components in nine type-A populations of 3,947 F2 lines. c, Gene structure of TAC1 and tac1 alleles. d, The performances of tiller angle for three genotypes of the TAC1 gene in 438 F2 lines (mean ± s.e.). The three genotypes are colour-coded in d and e. e, Computational modelling of the planting number per unit area (mu, equivalent to one-fifteenth of a hectare) for TAC1 gene. f, The allele frequencies of hd3a and tac1 in 1,063 pairs of type-A parents and 254 pairs of type-B parents.

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The effects of hd3a on yield were weaker in early-season hybrids. This could be because the florigen gene hd3a is under the regulation of multiple floral repressors. As with homozygous early-flowering alleles of multiple genes (including Hd2, Ghd7 and Ghd8) in early-season hybrids, the hd3a allele delayed flowering only 4 days compared with the Hd3a allele, in marked contrast with the ~20 days delay seen in the background of middle-season hybrids. Correspondingly, hd3a did not show any significant improvement in grain yield on this genetic background.

TAC1 has been shown to regulate plant architecture in rice16. The tac1 allele, which has a mutation in the splicing site of the fourth intron, greatly improved plant architecture and allowed more efficient light capture (Fig. 3c). The heterozygous genotype (TAC1/tac1) could produce significantly more seeds (2.6% in grain number per panicle and 1.6% in thousand-seed weight) than the genotype of the male parental genotype (TAC1/TAC1). Because the tac1 allele reduced tiller number (by ~2.5%), grain yield per plant for the TAC1/tac1 was not significantly increased compared to TAC1/TAC1. However, the improved plant architecture seen in the TAC1/tac1 genotype results in the increase of ‘grain yield per unit area’. Compared with the genotype of male parents (TAC1/TAC1), the genotype of hybrids (TAC1/tac1) displayed a change of approximately 2°–3° in tiller angle, allowing an ~15% increase in planting density (Fig. 3d, e). We estimated that the heterozygous states at the two loci (hd3a and tac1) collectively resulted in significantly more seeds (overall 13.3% increase per panicle, P < 10−9) than the genotype of male parents, explaining ~73.5% of heterosis advantage (the percentage of differences between the hybrids and their male parents) in the type-A system.

We further investigated the allelic distribution for these candidate genes in the parental lines of 1,063 hybrid rice varieties of type A. Hd3a and TAC1 were mostly distributed in restorer lines (male parents of hybrids), whereas hd3a and tac1 were in CMS lines (female parents of hybrids) (Fig. 3f). In the 1,063 hybrids of type A, nearly all restorer lines (98.5%) had the ancestral type of Hd3a, whereas most CMS lines (76.7%) had the hd3a allele. Genome-wide measurement of the extent of divergence between restorer and CMS lines in type-A hybrids using the fixation index F-statistics (FST) enabled the identification of highly differentiated loci that included Hd3a and TAC1 (FST = 0.57 and FST = 0.33, respectively, higher than the whole-genome average of FST = 0.11). Other highly differentiated loci included two fertility restorer genes Rf3 (FST = 0.82) and Rf4 (FST = 0.40)17,18, and two yield-related genes Ghd7 (FST = 0.34) and LAX1 (FST = 0.63)19,20. In contrast to hd3a and tac1, the beneficial alleles of the other differentiated loci (for example, Ghd7 and LAX1) were contributed from the restorer lines. We also scanned the genomes of 254 type-B hybrids, in which we found that the heterosis-associated alleles of hd3a and tac1 were present in a small proportion (8% and 10%, respectively).

Notably, the hd3a allele in CMS lines may also have contributed to some reduced performance (Extended Data Fig. 5). Most conventional indica CMS lines carried the low grain-quality alleles of waxy and ALK21,22 and the susceptible allele of the Pi2/Pi9 locus for disease23. hd3a may also affect the introduction of wide-compatibility alleles of S5 for indica–japonica crosses24. Molecular-marker-assisted selection may be useful for improving the CMS lines.

LAX1 and Ghd8 in type-B system

The two-line system (type B), which included thermo-sensitive genic male sterility (TGMS) regulated by the gene tms5 and photoperiod-sensitive genic male sterility (PGMS) regulated by pms3, has greatly facilitated genetic improvements of maternal lines25,26. In the populations from TGMS hybrids, we identified two QTLs that included candidate genes Ghd8 and Tms5 with large effects on grain yield25,27. The alleles of ghd8 and tms5 in female parents corresponded to low production or sterility in the hybrid. Interestingly, the heterozygous state at both loci sometimes showed a marginal overdominance effect for plant height, seed-setting rate and grain yield per plant (Extended Data Fig. 6a, b). Although the two QTLs that included the candidate genes Ghd8 and Tms5 did sometimes show heterotic effects, we were uncertain whether single-gene overdominance or pseudo-overdominance (there could be another nearby gene contributing) was responsible for the heterosis. In one type-A population, we noticed that a QTL underlying several yield components was co-located with the local region around Tms5 (Extended Data Fig. 7a–c), whereas Tms5 itself was not the causal gene (both parents of the cross carried the wild-type allele Tms5, Extended Data Fig. 7d, e). We inferred that the overdominance phenomena observed at the Tms5 locus, as well as the Ghd8 locus, were probably due to pseudo-overdominance conferred by some tightly linked genes with beneficial alleles in female parents.

We identified several beneficial alleles of large effects contributed by female parents in the type-B system, including LAX1 for grain weight20, OsMADS51 for heading date28 and GW3p6 (a newly identified QTL) for grain weight, all with positive partial dominance. These candidate genes were commonly used genetic contributors that led to heterosis performance in type-B hybrids. Moreover, some type-B parental lines have special beneficial alleles, because the two-line hybrid system could provide broad genetic resources for the exploitation of heterosis. The variety ‘Liangyoupeijiu’ (LYP9) generated from a cross between PA64S with a minor background of japonica (for example, containing the japonica-type NAL1 allele)29 and 93–11 was an important PGMS hybrid variety. We found that the alleles contributed by PA64S at a number of loci could collectively explain ~89.2% of the heterosis advantage of LYP9 over 93–11 (Extended Data Fig. 6c and Extended Data Fig. 8). Similarly, in other type-B hybrids, the heterozygous status of only a small number of loci resulted in increased grain yield over the homozygous genotypes of male parents (Extended Data Fig. 6d and Extended Data Fig. 9).

Multiple beneficial alleles of type C

Owing to the identification and application in breeding of wide-compatibility alleles24, the development of intersubspecific hybrids has become feasible in the past few years. Although indica generally has better yield performance in Southern China, Southeast Asia and South Asia, japonica carries many beneficial alleles that are uncommon in indica gene pools (for example, nal1 and tac1)16,29. In each subspecies, there were several large-effect alleles (for example, Sd-1, Hd1, GW6a, OsGI, DEP1)30,31,32,33,34. Positive partial dominance and overdominance effects have served as the major causes of heterosis that make indica–japonica F1 hybrids produce much higher yields than their inbred parental lines (Fig. 4a). Two linked loci on the same chromosome in repulsion phase in two parental lines also tended to result in pseudo-overdominance (for example, GW6a in indica and Hd1 in japonica on chromosome 6; Sd1 in japonica and OsGI/LAX1 in indica on chromosome 1), similar to an example of heterosis for plant height in Sorghum6. In contrast to intrasubspecific hybrids, in which the female parents supplemented only a few key alleles, the female parents in the intersubspecific hybrids often contained many beneficial alleles.

Figure 4: Combination of multiple beneficial alleles in type-C hybrids.
figure 4

a, Plots of the index of middle-parent heterosis for each QTL underlying grain yield per plant (LOD > 3.5) against the physical positions. The alleles with better performance that were contributed from japonica and indica are coloured by blue and red, respectively. The loci that include previously reported candidate genes are highlighted in bold, and the newly identified QTLs were uniformly named according to the underlying yield component trait, chromosome and physical location. The asterisk indicates the genes with strong overdominance or pseudo-overdominance. b, The linkage mapping of grain number per panicle. LOD values are plotted against the physical positions, and the threshold (3.5) is indicated by a horizontal dashed line. c, The performances of grain yield per plant for three genotypes of the IPA1 gene in 1,037 F2 lines (**P < 0.001, two-tailed t-test).

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Plant architecture is a primary factor underlying ‘unit area yield’ in crops. As well as TAC1/tac1, we identified another candidate gene OsSPL14 (also named IPA1)35,36 for regulating plant architecture in an indica–japonica hybrid (Fig. 4b). The rare allele ipa1 strongly enhanced grain number per panicle, whereas the wild-type allele IPA1 performed better in seed-setting rate and panicle number. Heterozygous OsSPL14 (IPA1/ipa1) showed partial dominance for these yield traits (Extended Data Fig. 10). For yield per plant, the heterozygous state of OsSPL14 showed strong overdominance effect, producing higher grain yield than either IPA1/IPA1 or ipa1/ipa1 (P < 0.001, Fig. 4c), which could explain 48.1% of heterosis advantage in this hybrid cross. Comparing the sequence of ipa1 with that of IPA1, we found several variants in the promoter region but no differences in the coding regions, suggesting that the strong overdominance in yield was related to gene expression in the heterozygous state of OsSPL14. These candidate genes for plant architecture (including OsSPL14 and TAC1) may be relevant for understanding balance between early growth and late growth rates and between yield per plant and planting density.

Conclusions

In summary, integrated genomic analyses in large samples have provided insights into the genetic basis of heterosis in rice. Our studies suggest that there are a few loci from female parents that contribute to heterosis within each subgroup, but that these are not universally shared across our representative elite rice lines. The heterozygous state of these loci mostly acted through the way of positive partial dominance, and some loci had to be heterozygous in breeding owing to pseudo-overdominance or ‘overdominance’ in overall performance. The heterotic effects of these loci in rice suggests that additive, partial dominance and overdominance may have been observed owing to allelic dosage effects, consistent with the findings in maize37,38. Further studies are needed to resolve candidate genes contributing to heterosis for yield in rice, as well as their mechanisms.

These insights into the genomic architecture of grain yield in rice will be useful for further work to improve parental lines and create superior hybrids efficiently by optimizing the designs of breeding. Heterosis in maize (and also some other crops) from crosses between two genetically divergent groups may have similar genetic patterns with the intersubspecific hybrids in rice, awaiting further investigations through large-scale genomic approaches. We believe a better understanding of heterosis in crops will help in development of new strategies for meeting global food security.

Methods

No statistical methods were used to predetermine sample size. The experiments were not randomized and the investigators were not blinded to allocation during experiments and outcome assessment.

Sampling and sequencing

From the large collection of hybrid rice (n = 1,495) that had been sequenced and phenotyped, seventeen hybrid combinations were carefully selected to construct F2 populations, generating a total of 10,074 individual lines. The 17 combinations contained all major types of hybrid varieties in rice with a relatively high genetic diversity. The genomic DNA was extracted from the fresh leaf tissue of each F2 plant using the DNeasy Plant Mini Kit (Qiagen). The sequencing library was constructed with an insert size of 200–400 bp using TruePrep Tagment Enzyme (Tn5 transposase, Vazyme), which was then indexed with dual index primers (24 barcodes in one end and 16 barcodes in the other end for the paired-end reads) and amplified using TruePrep Amplify Enzyme39. The indexed DNA samples of 384 F2 lines with different barcodes were then mixed together in an equal molar concentration, followed with the purification by silica membrane column and the size-selection by agarose gel electrophoresis. Each mixture was loaded into one lane of the Illumina HiSeq2500 system. Overall, 10,074 individual lines was sequenced, generating 100-bp paired-end reads of ~1 terabase. Each F2 line had approximately 0.2× genome coverage. The paired-end reads of each F2 line were aligned against the reference genome sequence (IRGSP releases build 4.0 pseudomolecules of rice) individually using the package SMALT (version 0.5.7) with the parameters of ‘-i 700 -j 50 -m 50’. Only the reads mapped uniquely to the reference genome sequence were retained for SNP calling.

Genotype calling

Genotype calling was carried out in any chromosomal region covered by a given number of SNPs. The window size (the number of n consecutive SNPs in a window) was allowed to vary according to the marker density. For most F2 lines from indica–indica crosses, fifteen consecutive SNPs were used as a single window. We applied a sliding window approach to evaluate the raw calls of multiple SNPs and performed the genotype calling using a Bayesian-based algorithm14. The window slid along a chromosome at a step size of one SNP, until all the genotypes were called for the entire chromosome. The recombination map was constructed for each chromosome in 10,074 F2 lines. Detailed procedures in genotyping, recombination breakpoint determination and map construction were implemented in the pipeline SEG-Map with default parameters14.

Phenotyping and linkage analysis

The F2 seeds from the 17 populations were germinated and planted in the experimental fields in Hangzhou, China (at N 30.32, E 120.12) in summer of 2014 from May to October. The phenotyping for this work involved a wide range of grain-yield traits in the field. All the F2 lines were grown in the consecutive farmland with well-distributed soil status in uniform condition with low use of fertilizers. Some hybrids and the restorer lines were also planted to estimate the yield advantage. Heading date was recorded daily as the number of days from sowing to the observation of first inflorescences that emerged above the flag leaf sheath. Plant height was measured from the soil surface to the apex of the tallest panicle. Tiller angle was measured between tiller and the ground level when panicles fully emerged. The yield-related traits, including panicle number, grain number per panicle, seed-setting rate and grain weight per 1,000 grains, were measured in the laboratory following harvest. Grain weight was initially obtained by weighing ~600 fully filled grains, which was then converted to a 1,000-grain weight value. The panicle number and the spikelets produced per panicle were counted manually. In each F2 population, QTL analysis of the yield traits was performed by the composite interval mapping (CIM) method implemented in the software Windows QTL Cartogapher (version 2.5)40 with the cross type of SF2, a window size of 10-cM and a step size of 2 cM. The LOD values were determined on the basis of likelihood ratio tests under a hypothesis allowing both additive and dominance effects. QTLs with LOD value higher than 3.5 were called, of which the physical location was retrieved from the peak location. The newly identified QTLs were uniformly named according to the major trait, chromosome and physical location (for example, GW3p6 represented a QTL for grain weight in the short arm of chromosome 3 within the genomic loci of 6–7 Mb).

GWAS and population genetics analysis

Integrating the ultra-dense genotype map of each F2 population and the haplotype map of their parental lines, the genotype calls were assigned at 1.7 million common SNPs for all the 10,074 F2 line. Principal component analysis of the SNPs was performed using the software EIGENSOFT (version 5.0.1)41, and the first two principal components were used in the analysis of the genetic structure of the 17 populations. Using the combined genotypic information, association analysis was conducted by the mixed model to correct for pedigree relatedness using EMMAX software package42. After filtering the SNPs with high missing data rate (20%) and low minor-allele frequency (2%), a total of 1,482,139 SNPs were used in final associations. The matrix of simple matching coefficients was used to build up the kinship matrix (10,074 × 10,074) for the correction. We defined a whole-genome significance threshold (1 × 10−6) according to permutation tests.

Previously we constructed a genome map for 1,495 elite hybrid rice varieties and their inbred parental lines through whole-genome sequencing13. The whole-genome data of the parental lines of 1,063 hybrid rice varieties of type A and 254 of type B were re-analysed for scanning highly differentiated loci between the group of restorer lines and that of CMS lines. The differentiation statistics (FST) were computed in each 100-kb window across 12 rice chromosomes. For 1,495 elite hybrid rice varieties, we carried out GWAS for heading date, grain number, colouration, grain quality and disease-resistance traits in the previous publication13. The lead SNPs for Hd3a, ALK, Waxy, OsC1, Pi2/Pi9 and TAC1 from previous GWAS13,43 were used to calculate their allele frequencies in the 1,063 pairs of restorer lines and CMS lines in type A.

Allelic variation detection for identified QTLs

In order to identify allelic variation (especially SNPs and indels in coding regions) for the identified QTLs in the large populations, all the Illumina reads were mapped to the reference rice genome with BWA (version 0.7.1)44 using default parameters to generate the BAM files. For the mapped reads, PCR duplicates were removed by the ‘MarkDuplicates’ module in Picard package (version 1.119). To reduce miscalls caused by misalignment in highly polymorphic regions, we realigned the reads at the target loci with the ‘IndelRealigner’ function in the GenomeAnalysisTK (version 3.4.0)45. Taking all of the realigned BAM results together, the sequence variants of high quality were called using ‘UnifiedGenotyper’ in the GenomeAnalysisTK and filtered by the base depths (≥5 at least). Effects of the sequence variants were evaluated according to the gene models of Nipponbare in the RAP-DB (release 2) across the rice genome, with manual checks for the coding structures of the well-characterized genes in rice. The candidate genes from GWAS and linkage analysis were identified based on both the genetic mapping information and the allelic mutations observed from the sequence reads. Using the comprehensive dataset, the allelic variation and their combinations for the key genes (including Hd3a, TAC1, Ehd1, LAX1, Tms5 and OsSPL14) were examined across the 17 populations.

Allele effect estimation

After the QTLs were identified, allele effect was first assessed through a comparison of both homozygous genotypes. The average phenotypic measurements of heterozygous genotypes and homozygous genotypes were further calculated for the estimations of the index of dominance-effect/additive-effect (d/a) and the index of middle parent heterosis for each QTL. Two-tailed t-tests were used to evaluate the significance of the phenotypic contribution of the heterozygous genotypes versus the genotypes of male parents at the key loci, including Hd3a and TAC1. The measurements with a seed-setting rate <20% and grain yield per plant <5 g were excluded from the comparisons to avoid the influence of sterile traits. For the TAC1 gene underlying tiller angle, the planting number per unit area was modelled using the formula n = C/(cosα)2, where n is the plant number per mu (one fifteenth of a hectare), C was an empirical constant (given as 2,500 here for typical inbred indica lines) and α was the tiller angle measured between the tiller and the ground level.

Genetic modelling for heading date

For modelling the genetic architecture of heading date, multiple linear regressions were performed to examine the effects of diverse allelic combinations in four major loci (Hd3a, Ghd7, Hd2 and Ehd1) using the ‘proc reg’ procedure in SAS. The late-flowering alleles of Hd3a and Ehd1 came from female parents and those of Ghd7 and Hd2 were from male parents. Before fitting the model, each marker was recorded. The value 0 was used for the allele from male parents, the value 1 was used for that from female parents and the value for heterozygous genotype depended on the dominance effect of each gene. In the modelling procedure, we only used the genotypic data at the four loci and the phenotypic data for heading date in one type-A population (Guanghui998 × Tianfeng), which resulted in the estimates for recoding, weighting and the intercept. The estimates from this training population were used for the prediction of heading date in the remaining eight populations. The value R was calculated from the correlation between the predicted values from modelling and the observed phenotypic data for heading date in eight populations.